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MTS-2011-Sheet1.dvi
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2010-2011/MTS-2011-Sheet1.pdf6 May 2011: c) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... Prove the following:. (a) G is homogeneous: given any x, y G there is a homeomorphism φ : G G such thatφ(x) = y. -
MTS-2010-Sheet1.dvi
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2009-2010/MTS-2010-Sheet1.pdf29 Apr 2010: c) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... Prove the following:. (a) G is homogeneous: given any x, y G there is a homeomorphism φ : G G such thatφ(x) = y. -
ON MODULAR UNITS A. J. Scholl Introduction. In [6], ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/u.pdf29 Jan 2010: The irreducibility of Sp(χ) for such a prime pin conjunction with a) implies that Φq,χ = 0. ... Since GẐ acts transitively on |MK |,. there exists a positive constant cn Q such that for every x |M|,. -
ANALYSIS II—EXAMPLES 4 Mich. 2014 The questions marked with ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet4revised.pdf27 Feb 2015: Does there exist c (a,b)such tat f(b) f(a) = f′(c)(ba)?(v) Let f: Rn Rn be a C1 map with ‖Df(x)I‖ 1/2 for ... continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). -
1 Metric & Topological spaces, Sheet 1: 2006 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2005-2006/2006sheet1.pdf4 May 2006: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... a) Prove (R, TS ) is Hausdorff. (b) Prove (R, TS ) is not metrisable. -
TECHNIQUES IN COMBINATORICS: EXAMPLES SHEET 2 Michaelmas Term 2014 ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TechniquesinCombinatorics/techniques2014examples2.pdf20 Oct 2014: Prove that there are. subsets A X and B G such that |η(A,B) δαβ| θ, where θ > 0 depends on γ andδ only. ... x1,. ,xk such that no xi xj with i,j distinct is a quadratic residue mod p. -
Applications of Mirror Symmetry
https://www.dpmms.cam.ac.uk/~mg475/riverside.pdf16 Jul 2014: singularities is a real manifold Bwith an open subset B0 B, with := B B0 of codimension 2, such that B0 has the structure of an integral affine manifold.We also allow ... X Spec k[[t]]. such that the special fibre satisfies. X0 =. σPmax. Pσ. and the -
Solving cubic equations
https://www.dpmms.cam.ac.uk/~wtg10/cubic.html25 Apr 2002: 3. dy. 2. eyf in y and finally (the most important part) cleverly choose u and v in such a way that d. ... d. 2. equal to 3e, or it may be that, although such a choice exists, the dependence of u and v on a, b and c is so complicated that we -
The philosophy of mathematics
https://www.dpmms.cam.ac.uk/~wtg10/philosophy.html20 Sep 2004: And yet there doesn't seem to be such a fact of the matter. ... Similarly, if B={x:Q(x)} then the statement "A cap B = emptyset" means nothing more than that there is no x such that P(x) and Q(x). -
Michaelmas Term 2023 Number Theory: Example Sheet 1 of ...
https://www.dpmms.cam.ac.uk/study/II/NumberTheory/2023-2024/number_theory-23-1.pdf27 Oct 2023: Calculate d = (a, b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a, b) c log b d. -
Duality∗ Ernst Specker (translated by Thomas Forster) June 14, ...
https://www.dpmms.cam.ac.uk/~tef10/dualityquinevolume.pdf14 Jun 2018: an〉 such that for all i < n 1, ai 6= ai1, and that〈a0,. ... C0 I is symmetrical. C1 For two distinct elements a and b there is a unique c such that I(a,c) andI(b,c). -
The fundamental theorem of arithmetic not always true.
https://www.dpmms.cam.ac.uk/~wtg10/meta.fta.html11 May 2001: 1/2. , so two such numbers must multiply to something of modulus at least 5. ... Analogues of Euclid's algorithm work in other contexts, such as the ring of all polynomials with rational coefficients. -
Cahiers du Centre de logiqueVolume 17 The Paris-Harrington Theoremin…
https://www.dpmms.cam.ac.uk/~tef10/parisharringtontalk.pdf5 Nov 2009: Then there is a subset B A of size β such that all theunordered γ-tuples from it are in the same piece of the partition. ... b) the property of relative largeness which plays such a central rôle inParis -Harrington is ill-typed in at least two senses. -
TOPICS IN SET THEORY: Example Sheet 3 1 Department ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TopicsinSetTheory/TST%20Example%20Sheet%203%202014.pdf19 Dec 2014: if B = {min(Xα) : α < κ} is stationary.(iii) If {Xα : α < κ} is pairwise disjoint, then there exists a [κ]κ such that. ... Prove that there exists a family F of2κ stationary subsets of S such that if A 6= B F , then A B and B A arestationary in -
TOPICS IN SET THEORY: Example Sheet 1 1 Department ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TopicsinSetTheory/TST%20Example%20Sheet%201%202014.pdf19 Dec 2014: i) Prove that 20 = 1 implies the following assertion SD :there exist sets A and B such that R2 = AB and Al and Bm are countablefor every horizontal line l and ... exist x1 and x2 such that x1 6 f(x2) and x2 6 f(x1). -
Continuity and open sets II
https://www.dpmms.cam.ac.uk/~wtg10/easyanalysis2.html7 Aug 2002: A formal statement of the result to be proved. Let X and Y be metric spaces and let f be a function from X to Y such that f. ... in U (for the reason I have just given) and such that [f(y) is in U] implies [d(f(x),f(y)) < a]. -
Commutativity of Multiplication
https://www.dpmms.cam.ac.uk/~wtg10/commutative.html25 Nov 2002: We also know that CxB is a subset of AxB, and that the set-theoretic difference AxB-CxB consists of all ordered pairs (m,x) such that x is in B. ... me that if m> n then there is a unique positive integer t such that nt=m. -
Games played on an illfounded membership relation Thomas Forster ...
https://www.dpmms.cam.ac.uk/~tef10/charlestalk.pdf9 Apr 2012: Obviously we want I and II be the least fixedpoints for these two operations, and since P and b are both monotone functionson the complete poset 〈V,〉 there will be such ... More generally, butless idiomatically, we can write:. There is no -least set -
Analysis II Example Sheet 1 Michaelmas 2006 MJW Corrections ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2006-2007/ex1.pdf13 Oct 2006: b) Suppose that f is bounded. Prove that there exists M and n0 such that |fn(x)| < M for all. ... a,b] such that fn(xn) does not converge to f(x). 7. Which of the following functions f are (a) uniformly continuous, (b) bounded on [0,)? -
A semantic characterization of the well-typed formulæ of λ-calculus…
https://www.dpmms.cam.ac.uk/~tef10/many.tcs.pdf29 Mar 2007: The intuitionistic correctness of wffs such as. 4. A (B C). ... B (A C) reveals an infinite family of terms such as.
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